Breadcrumbs

You are here:

Adaptive Low-Rank Methods for Problems on Sobolev Spaces with Error Control in L 2

AUTHORS

Markus Bachmayr, Wolfgang Dahmen

ABSTRACT

Low-rank tensor methods for the approximate solution of second-order elliptic partial differential equations in high dimensions have recently attracted significant attention. A
critical issue is to rigorously bound the error of such approximations, not with respect to a fixed
finite dimensional discrete background problem, but with respect to the exact solution of the
continuous problem. While the energy norm offers a natural error measure corresponding to the
underlying operator considered as an isomorphism from the energy space onto its dual, this norm
requires a careful treatment in its inter- play with the tensor structure of the problem. In this
paper we build on our previous work on energy norm-convergent subspace-based tensor schemes
contriving, however, a modified formulation which now enforces convergence only in L2 . In order
to still be able to exploit the mapping properties of elliptic operators, a crucial ingredient of
our approach is the development and analysis of a suitable asymmetric precondition- ing scheme. We
provide estimates for the computational complexity of the resulting method in terms of the solution
error and study the practical performance of the scheme in numerical experiments. In both regards,
we find that controlling solution errors in this weaker norm leads to substantial simplifications
and to a reduction of the actual numerical work required for a certain error tolerance.